Determine the annulus of convergence of the Laurent series and find the sum fo the series there.

Σ n(-1)^n 2^ -│n│Z^n
( sum of n = neg. infinity to infinity)

Thanks

You have,
Necessay and Sufficient (slip the sums),+
You can get rid of the absolute value because it takes on positives and negatives,+
In the first summand, let (if then it converges, nothing is lost).
Thus, we can write,
The non-alternating term is,.
Use the generalized ratio test,.
Hence the reciprocal of this, which is 2, is the radius of absolute convergence.
Thus, (in both series)
Take reciprocal of both sides on the second one,
Thus,
This is an annulus of absolute convergence.
I still think you need to check the boundary for convergence.

These two series can be summed as they are the derivatives of geometric series which converge for and respectively, and have singularities on the circles and respectively, which is sufficient to conclude that the annulus of convergence is .

These two series can be summed as they are the derivatives of geometric series which converge for and respectively, and have singularities on the circles and respectively, which is sufficient to conclude that the annulus of convergence is .

RonL

I was thinking about something else, these are alternating, decreasing sequences to zero. Thus, can we use the Leibniz alternating series test? I was not certain about that because this was a complex analysis type question.